It is well-known that the initial value problem for (REF ) is globally well-posed in
\(L^1 (rdrdz)\cap L^\infty (\Sigma )\) , see [1]}, [2]}, [3]}.
A bounded solution \(\omega \)  to (REF ) with sufficient space
decay actually satisfies
\(t\mapsto \int _{\mathbb {R}^2} r \omega (r,z,t) \,drdz = constant, \quad t\mapsto \Vert w(\cdot , t)\Vert _{L^\infty (\mathbb {R}^2)} = constant.\) 
